Abbreviation: ActLat

An \emph{action lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,0,\cdot,1,^*,\backslash ,/\rangle$ of type $\langle 2,2,0,2,0,1,2,2\rangle$ such that

$\langle A,\vee,0,\cdot,1,^*\rangle$ is a Kleene algebra

$\langle A,\vee,\wedge\rangle$ is a lattice

$\backslash$ is the left residual of $\cdot $: $y\leq x\backslash z\Longleftrightarrow xy\leq z$

$/$ is the right residual of $\cdot$: $x\leq z/y\Longleftrightarrow xy\leq z$

Let $\mathbf{A}$ and $\mathbf{B}$ be action lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(x^*)=h(x)^*$, $h(0)=0$, $h(1)=1$

Example 1:

$\begin{array}{lr} f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &20
f(5)= &149
f(6)= &1488
\end{array}$

Commutative action lattices

Action algebras

Residuated lattices